Scheduling Problems with Constrained Rejections
Sami Davies, Venkatesan Guruswami, Xuandi Ren
TL;DR
We study bicriteria scheduling variants with constrained rejections for Makespan on Unrelated Machines and the Santa Claus problem, formalizing the trade-off between the fraction of scheduled jobs/agents and the corresponding makespan or minimum value. We provide a $0.6533$-fraction randomized algorithm achieving a makespan of $1.5$ times the optimal $T$ by combining a large-edge matching, configuration-LP rounding, and a small-edge greedy procedure. We also establish NP-hardness for the Santa Claus bicriteria objective and introduce bicriteria Set Packing, presenting both algorithmic results and hardness via a CSP-based gadget. The results illuminate the shape of the trade-off curves and motivate future work on transition points and thresholds in these bicriteria settings.
Abstract
We study bicriteria versions of Makespan Minimization on Unrelated Machines and Santa Claus by allowing a constrained number of rejections. Given an instance of Makespan Minimization on Unrelated Machines where the optimal makespan for scheduling $n$ jobs on $m$ unrelated machines is $T$, (Feige and Vondrák, 2006) gave an algorithm that schedules a $(1-1/e+10^{-180})$ fraction of jobs in time $T$. We show the ratio can be improved to $0.6533>1-1/e+0.02$ if we allow makespan $3T/2$. To the best our knowledge, this is the first result examining the tradeoff between makespan and the fraction of scheduled jobs when the makespan is not $T$ or $2T$. For the Santa Claus problem (the Max-Min version of Makespan Minimization), the analogous bicriteria objective was studied by (Golovin, 2005), who gave an algorithm providing an allocation so a $(1-1/k)$ fraction of agents receive value at least $T/k$, for any $k \in \mathbb{Z}^+$ and $T$ being the optimal minimum value every agent can receive. We provide the first hardness result by showing there are constants $δ,\varepsilon>0$ such that it is NP-hard to find an allocation where a $(1-δ)$ fraction of agents receive value at least $(1-\varepsilon) T$. To prove this hardness result, we introduce a bicriteria version of Set Packing, which may be of independent interest, and prove some algorithmic and hardness results for it. Overall, we believe these bicriteria scheduling problems warrant further study as they provide an interesting lens to understand how robust the difficulty of the original optimization goal might be.